Learning in Riemannian Orbifolds

Statistical data analysis and learning in Riemannian orbifolds is motivated by applications, where the data we want to learn on are naturally represented by finite combinatorial structures such as point patterns, trees, and graphs. Examples from structural pattern recognition that learn on structured data include estimating central points of a distribution on graphs such as the mean and median [9, 16, 15, 21], central clustering of graphs [10, 12, 13, 14, 19, 15, 23], learning graph quantization [17], and multilayer perceptrons for graphs [20]. In retrospect, the structure space framework proposed by [18] theoretically justifies the above approaches in the sense that they actually minimize an empirical risk function on structures. Since minimizing an empirical risk function is usually computationally intractable, the ultimate challenge consists in constructing efficient algorithms which are capable to return optimal or at least suboptimal solutions. From the point of view of statistical pattern recognition, however, the ultimate goal is not to determine a good solution of an empirical risk function, but rather to discover the true but unknown structure of the data with respect to its distribution.